hatchet vs claw hatchet

hatchet and claw hatchet both are nouns.

Word	Noun	Adjective	Verb	Adverb
hatchet	Yes	No	No	No
claw hatchet	Yes	No	No	No

As nouns, claw hatchet is a hyponym of hatchet; that is, claw hatchet is a word with a more specific, narrower meaning than hatchet:

hatchet: a small ax with a short handle used with one hand (usually to chop wood)
claw hatchet: a hatchet that has a cleft for pulling nails

Other hyponyms of hatchet include broad hatchet, half hatchet.

hatchet (noun)	claw hatchet (noun)
a small ax with a short handle used with one hand (usually to chop wood)	a hatchet that has a cleft for pulling nails
weapon consisting of a fighting ax; used by North American Indians